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<h3 class="heading"><span class="type">Paragraph</span></h3>
<p>Consider the equation</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_13.html ./knowl/eq2_14.html ./knowl/eq2_14.html">
\begin{equation*}
\frac{\textrm{d} y}{\textrm{d} x}=f(x, y).
\end{equation*}
</div>
<p class="continuation">We write</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_13.html ./knowl/eq2_14.html ./knowl/eq2_14.html">
\begin{equation*}
f(x, y)=-\frac{M(x, y)}{N(x, y)}.
\end{equation*}
</div>
<p class="continuation">Then, we have</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_13.html ./knowl/eq2_14.html ./knowl/eq2_14.html">
\begin{equation*}
M(x, y)+N(x, y)\frac{\textrm{d} y}{\textrm{d} x}=0.
\end{equation*}
</div>
<p class="continuation">In this section, we consider the special case that <span class="process-math">\(M(x, y)=M(x), \quad N(x, y)=N(y)\text{,}\)</span> i.e., <span class="process-math">\(f(x, y)=\frac{-M(x)}{N(y)}\text{.}\)</span> In this case, the ODE becomes</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_13.html ./knowl/eq2_14.html ./knowl/eq2_14.html">
\begin{equation}
M(x)+N(y) \frac{\textrm{d} y}{\textrm{d} x}=0,\tag{2.3.1}
\end{equation}
</div>
<p class="continuation">or it can be written as</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_13.html ./knowl/eq2_14.html ./knowl/eq2_14.html">
\begin{equation}
M(x) \textrm{d} x=-N(y) \textrm{d} y.\tag{2.3.2}
\end{equation}
</div>
<p class="continuation">Equations (<a href="" class="xref" data-knowl="./knowl/eq2_13.html" title="Equation 2.1.13">(2.1.13)</a>), (<a href="" class="xref" data-knowl="./knowl/eq2_14.html" title="Equation 2.3.2">(2.3.2)</a>) are separable ODEs. And further integration of (<a href="" class="xref" data-knowl="./knowl/eq2_14.html" title="Equation 2.3.2">(2.3.2)</a>) gives</p>
<div class="displaymath process-math" data-contains-math-knowls="./knowl/eq2_13.html ./knowl/eq2_14.html ./knowl/eq2_14.html">
\begin{equation*}
\int M(x) \textrm{d} x=-\int N(y) \textrm{d} y+C.
\end{equation*}
</div>
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